A Theory of Production, Matching, and Distribution∗

Sephorah Mangin†

April 2015

Abstract

This paper presents a unified approach to production, matching, and distributionin an environment with labor market frictions. I use competing auctions as both awage determination mechanism and a microfoundation for an aggregate production andmatching technology. For any well-behaved distribution of firm productivities, I showthat the aggregate production function exhibits standard neoclassical properties andan elasticity of substitution between capital and labor that is always less than or equalto one. If the distribution is Pareto or power law, this function is particularly tractable.In general, factor income shares vary with the degree of competition between firms tohire workers, the value of non-market activity, and characteristics of the underlying firmproductivity distribution. Unlike Diamond-Mortensen-Pissarides style search modelswith Nash bargaining, production and distribution are endogenously tightly connected:the economy satisfies a generalized version of the Hosios condition.

JEL Codes: J64, E23, E24, E25

Keywords: Aggregate production function, elasticity of substitution, factor shares,Pareto distribution, power law, extreme value theory, competing auctions, directedsearch, competitive search, Hosios condition

∗I would especially like to thank Chris Edmond, Ian King, Ricardo Lagos, and Rob Shimer for usefulfeedback and advice. I would also like to thank numerous individuals for helpful discussions as well asseminar participants at the University of Edinburgh, the Capital Theory Working Group at the Universityof Chicago, the Chicago Fed, the Philadelphia Fed, the St Louis Fed, the University of Wisconsin-Madison,Melbourne, Adelaide, Queensland, Monash, ANU, and Deakin; and the SED Annual Meeting 2012, theSearch and Matching Annual Meeting 2012, the MEA Meeting 2012, the Midwest Macro Meeting 2012, theWorkshop on Macroeconomic Dynamics 2013, and the North American Winter Meeting of the EconometricSociety 2014. I am grateful to Richard Paris for his assistance in the proof of Lemma 4. I also thank theBecker Friedman Institute for its hospitality.†Department of Economics, Monash University. Email: sephorah.mangin@monash.edu.

I Introduction

The standard neoclassical approach to production and distribution is perhaps best

represented by Cobb and Douglas (1928), which introduces the Cobb-Douglas aggre-

gate production function and presents evidence that the "processes of distribution"

approximate the "laws of production". Essentially, this means factors are paid their

marginal products and the production technology determines the factor income dis-

tribution: labor’s income share equals its output elasticity. For the Cobb-Douglas

function, the tight connection between production and distribution is particularly

clear since this elasticity is a parameter and factor shares are constant.

Search-theoretic models of the labor market feature a frictional matching process

between workers and firms that gives rise to unemployment. In Diamond-Mortensen-

Pissarides (DMP) models with random matching and generalized Nash bargaining,

the distribution of output is governed by the bargaining parameter. This breaks

the tight connection between production and distribution: these models are not con-

strained effi cient unless the Hosios condition is arbitrarily imposed.1

This paper attempts to bridge the gap between the search-theoretic approach

and the standard neoclassical paradigm by providing joint microfoundations for pro-

duction and distribution in a frictional labor market environment. The framework

delivers a unified aggregate production and matching function that incorporates la-

bor market frictions and hence unemployment. Unlike DMP style models, production

and distribution are tightly connected and the economy is constrained effi cient.

For any well-behaved distribution of firm productivities, I show that the resulting

constant-returns-to-scale aggregate production function exhibits standard neoclassi-

cal properties: it is increasing and strictly concave, and it satisfies a "weak" version

of the Inada conditions.2 Moreover, I prove that the aggregate elasticity of substitu-

tion between capital and labor is always less than or equal to one. This theoretical

result contributes to the current debate in macroeconomics regarding the value of this

elasticity and complements the large body of empirical work, such as Antras (2004)

and Oberfield and Raval (2014), that suggests this elasticity is below one.3

1The Hosios condition states that constrained effi ciency holds when labor’s bargaining powerequals its elasticity in the matching function. See Hosios (1990).

2By "weak", I mean that all of the Inada conditions are satisfied except limθ→0 f ′(θ) is finite. Asdiscussed in Section II, however, the condition that limθ→0 f ′(θ) = +∞ is stronger than necessary.

3See also Chirinko (2008). As discussed in Rognlie (2014), almost all estimates in the literatureare below one. Karabarbounis and Neiman (2014) is a recent exception.

1

I use a competing auctions framework similar to Peters and Severinov (1997).4

The environment can be seen as a generalization of competitive search that incorpo-

rates both many-on-one meetings and private information. Workers ("sellers") post

second-price auctions with reservation wages ("reserve prices").5 Firms ("buyers")

pay a fixed cost to hire a unit of capital and approach workers at random. After

approaching a worker, firms draw private match-specific productivities independently

from a common distribution. A firm’s productivity represents its "valuation" of a sin-

gle unit of labor. The highest productivity firm targeting a worker hires that worker,

and his wage equals the second-highest productivity (or his reservation wage in a

one-on-one meeting). If no firms approach, the worker is unemployed.

Competition between firms to hire workers performs two key functions simulta-

neously. First, it endogenizes output per match and hence aggregate output, which

depends on both the labor market tightness and the firm productivity distribution.

Second, it endogenizes the distribution of output between workers and firms by en-

dogenizing workers’relative "bargaining" position.

Greater competition between firms to hire workers decreases capital’s share. Equiv-

alently, labor’s share is increasing in the labor market tightness. The equilibrium labor

share is increasing in the value of non-market activity, which can be interpreted as a

simple proxy for various labor market institutions such as unemployment insurance.

In addition, labor’s share is influenced by the nature of the underlying firm pro-

ductivity distribution. These predictions contrast markedly with the view found in

Piketty (2014) that capital’s share may increase indefinitely over time due to capital

accumulation and an elasticity of substitution greater than one.

While all of the key results in this paper hold for any well-behaved distribution

of firm productivities, I focus on the Pareto distribution as a lead example. In this

case, the aggregate production function is quite tractable. In general, the elasticity

of substitution is below one, but in the limiting case where unemployment disappears

this function is asymptotically Cobb-Douglas. This result is related to —but distinct

from —Houthakker (1955), Jones (2005), and Lagos (2006).6 Alternatively, in the

special case where the firm productivity distribution is degenerate, aggregate output

is given by a standard urn-ball matching function. Importantly, while both the Cobb-

4Peters and Severinov (1997) builds on Wolinsky (1988) and McAfee (1993).5McAfee (1993) proved that if sellers can choose mechanisms, it is an equilibrium outcome for

sellers to hold identical auctions with effi cient reserve prices and for buyers to randomize over sellers.6See the discussion under Related Literature.

2

Douglas production function and the urn-ball matching function can be isolated as

special cases or limits, neither of these is the focus of this paper. Instead, the unique

focus of this paper is on the unified aggregate production and matching technology

that emerges in this frictional labor market environment.

In general, factor shares are variable. However, constant factor shares can arise

in two distinct cases. The first is the limiting case where unemployment goes to zero:

for any well-behaved distribution, factor shares are asymptotically constant. More

interesting is the second case, which holds for finite labor market tightness ratios.

Constant factor shares arise when the firm productivity distribution is Pareto and

the value of non-market activity equals the minimum firm productivity. While the

connection between the Pareto distribution and the Cobb-Douglas function is well-

known since Houthakker (1955), this result is novel. When wages are determined by

competitive bidding to hire workers, the Pareto distribution can potentially generate

constant factor shares even when the production function is not Cobb-Douglas.

I provide a generalization of the Hosios condition that links production, matching,

and distribution in a natural and intuitive way. If workers’outside option is zero,

this condition is simple: constrained effi ciency obtains when labor’s share equals its

elasticity in the aggregate production function. In the present model, the general

version of this condition holds endogenously. Since the effi ciency of entry in compet-

ing auctions is well-known, this is not surprising. However, the generalized Hosios

condition is useful because of its simplicity and breadth of scope.

Related Literature. While a number of papers have provided microfoundations

for an aggregate matching function in search environments,7 Lagos (2006) is the only

other paper —to my knowledge —that provides microfoundations for a tractable aggre-

gate production function in a search-theoretic model. In that sense, my paper is very

closely related in spirit to Lagos’. While similar in spirit, my contribution is different.

Lagos uses a DMP style model with an exogenous matching function and generalized

Nash bargaining, while my approach uses competing auctions. This difference has

important consequences. Since Lagos derives a Cobb-Douglas aggregate production

function, there is indeed a sense in which his paper reconciles the search-theoretic

and neoclassical paradigms. However, the tight connection between production and

7In addition to the directed search literature discussed in this section, see Lagos (2000), Stevens(2007), Shimer (2007), Sattinger (2010), and Kohlbrecher, Merkl, and Nordmeier (2014).

3

distribution remains broken: the parameter governing the production technology is

different to the bargaining parameter and constrained effi ciency does not hold.

This paper is also closely related to Jones (2005), as well as Houthakker (1955)

who originally derived an exact Cobb-Douglas aggregate production function using

Leontief local production technologies and a generalized Pareto type distribution of

capacities.8 There are some important differences between my approach and the re-

sults of Jones (2005) and Houthakker (1955). First, labor market frictions are central

to the present paper. Second, I develop joint microfoundations for both production

and distribution by modelling wage determination explicitly. Third, I provide general

results that hold for any well-behaved distribution of firm productivities, not just the

Pareto. Finally, even for the Pareto distribution, my results are different: the aggre-

gate production function is not generally Cobb-Douglas when we incorporate labor

market frictions. Instead, this function is asymptotically Cobb-Douglas only in the

special limiting case where unemployment disappears.

My limiting aggregation result arises from the fact that the endogenous distrib-

ution of output across workers converges to a Fréchet extreme value distribution as

the labor market tightness ratio becomes large. In addition to Jones (2005), the close

connection between this paper’s limiting results and extreme value theory links it

mathematically to papers such as Kortum (1997), Eaton and Kortum (1999, 2002),

Gabaix and Landier (2008), and Oberfield (2013). In particular, this paper’s general

result regarding the limiting behavior of factor income shares is similar to an inde-

pendent result by Gabaix, Laibson, Li, Li, Resnick, and de Vries (2014) regarding the

limiting value of markups when the number of firms is large. My paper’s focus on la-

bor market frictions distinguishes my contribution from this existing literature. Since

unemployment is crucial in a labor market context, I focus on finite labor market

tightness ratios and not on the asymptotic results, although I do discuss these.

This paper is closely linked to the competing auctions literature, both the early

work of Peters and Severinov (1997) and the recent contribution of Albrecht, Gautier,

and Vroman (2014), who derive effi ciency results in a competing auctions environ-

ment with heterogeneous sellers. This paper is also related to the wide literature on

directed and competitive search. Unlike DMP models, constrained effi ciency gener-

8Levhari (1968) shows that Houthakker’s result can be generalized to a CES production functionwith σ < 1 using a beta distribution. Dupuy (2012) uses a tasks assignment model similar to Rosen(1978) to derive a CES aggregate production function.

4

ally holds in this class of models. Moen (1997) and Shimer (1996) first established

constrained effi ciency in models of this kind. Other early contributions include Mont-

gomery (1991), Peters (1991), Acemoglu and Shimer (1999), Julien, Kennes, and

King (2000), Burdett, Shi, and Wright (2001), Shi (2001), and Mortensen and Wright

(2002).9 Guerrieri (2008) and Moen and Rosén (2011) introduce match-specific pri-

vate information into competitive search economies with one-on-one meetings, while

Guerrieri, Shimer, and Wright (2010) consider ex ante heterogeneity.

Julien et al. (2000) model workers as "sellers" of labor and firms as "buyers" in

a directed search environment where all firms have the same productivity and there

is no private information. The large economy version of Julien et al. (2000) can be

interpreted as a special case of the present model. My approach generalizes that

setting to one with endogenous output per match, a continuous wage distribution,

and a tractable aggregate production function. In fact, the framework in this paper

nests both directed search environments such as Julien et al. (2000) and competing

auctions models such as Peters and Severinov (1997) in the sense that I allow for the

coexistence of both informational and matching rents.

Shimer (2005) considers an economy similar to the present one, except that both

workers and firms are ex ante heterogeneous. Workers apply to firms and firms

hire the most productive applicant. In a sense, Shimer’s framework is more general

since match output is a function of both worker and firm type. The present set-up

differs in at least two key respects. First, workers and firms are ex ante identical

and firms learn their productivities ex post, i.e. after approaching workers. Second, I

use a continuous distribution of firm productivities instead of a finite number of firm

types. This approach yields a tractable aggregate production function with standard

neoclassical features and elasticity of substitution below one.

Outline. Section II presents the model. Section III derives the aggregate production

and matching function and describes its properties. Section IV characterizes the

equilibrium. Section V examines distribution, in terms of both wages and factor

income shares. Section VI discusses effi ciency and Section VII concludes.

9More recent contributions to both directed and competitive search include Coles and Eeckhout(2003), Albrecht, Gautier, and Vroman (2006), Shi (2009), Galenianos and Kircher (2009, 2012),and Menzio and Shi (2010, 2011). See also Postel-Vinay and Robin (2002).

5

II Model

There are two kinds of risk-neutral agents: workers and firms. There is a contin-

uum of ex ante identical workers of measure L and a continuum of ex ante identical

firms. The measure of firms who decide to enter is V and the ratio of such firms to

workers is θ ≡ V/L, the labor market tightness.

Workers are "sellers" of a single unit of labor and firms are "buyers". Workers

post second-price auctions and choose a reservation wage ("reserve price") to attract

firms.10 We focus on symmetric equilibria where all workers choose the same reser-

vation wage wR for a given market tightness θ.

Firms observe the reservation wage wR and, if they choose to enter, pay an entry

cost r to obtain one unit of capital. Total capital K is given by firms’ demand,

K = V, and hence θ = K/L, the capital-labor ratio. The cost r can be interpreted as

the rental rate of capital : it is the cost of hiring one unit of capital for a single period.

After paying the cost r, each firm approaches a single worker. Since firms are

uncoordinated and ex-ante identical, I assume that firms approach workers at random.

The actual number of firms approaching any given worker is a Poisson random variable

with parameter θ.11

A firm with productivity x can produce x units of output, with price normalized to

one, using a single unit of capital and a single unit of labor. Since labor is necessary

for production, a firm with productivity x is willing to pay up to x to purchase a

single unit of labor. Firms learn their productivities ("valuations") ex post. After

approaching a single worker, each firm draws a private match-specific productivity x

independently from a common distribution G.

Since it is a weakly dominant strategy for buyers to bid their true valuations in

second-price auctions, we assume that firms do so. A firm is successful in hiring

a worker if and only if it has the highest productivity for the particular worker it

approaches. Unsuccessful firms receive a payoff of zero.

If no firms approach a given worker, he is unemployed. By the Poisson distribution,

this occurs with probability e−θ, so u(θ) = e−θ is the unemployment rate. The

matching function that arises is urn-ball, namely m(θ) = 1− e−θ.12

10The main results of this paper also hold for first-price auctions due to revenue equivalence.11Since firms approach each worker with equal probability, the Poisson distribution arises because

we are taking the limit of a binomial distribution —a standard result.12The urn-ball matching function was first introduced by Butters (1977) and Hall (1977).

6

Workers who are unemployed receive the value of non-market activity, z ≥ 0. The

value of non-market activity z is a simple proxy for both the value of leisure and

various labor market institutions such as unemployment insurance.

If exactly one firm approaches a worker, it employs the worker and produces

output equal to its productivity x. The worker is paid his reservation wage wR in this

case. If two or more firms approach a worker, the firm with the highest productivity,

x1, employs the worker and produces output x1. The worker’s wage equals x2, the

second-highest productivity among the competing firms.

We restrict our attention to distributions that are well-behaved as defined below.

Definition 1. Let G be a cumulative distribution function that is differentiable with

density g = G′. Let εG(x) be the elasticity of 1−G, with respect to x,

(1) εG(x) ≡ xg(x)

1−G(x).

We say that G is well-behaved if and only if it has a finite mean, and support [xmin,∞)

where xmin ≥ 0, and εG(x) is weakly increasing, i.e. ε′G(x) ≥ 0.

The condition that ε′G(x) ≥ 0 is strictly milder than the increasing hazard rate

condition and it is satisfied by almost all standard distributions.13 For simplicity,

we normalize xmin = 1 and assume that z ≤ 1 so that workers always accept job

offers. To ensure there is positive firm entry, we also make the following assumption

regarding the distribution G.

Assumption 1. The distribution G satisfies EG(x) > z + r.

III Production

Matching and production are two aspects of a single process: both the employ-

ment status of a given worker and his expected output are determined by the number

of firms competing to hire the worker, which depends on the labor market tightness.

This process of matching and production generates an endogenous cross-sectional

distribution of output across workers, HG(x; θ). This distribution simultaneously in-

corporates two dimensions: the employment effect of the frictional matching process,

13Any distribution with an increasing hazard rate satisfies this condition, while the Pareto distri-bution is an example of a distribution with a decreasing hazard rate that still satisfies this condition.

7

which leads to unemployment for some workers; and the productivity effect of the com-

petition between firms, which leads to an allocation of labor towards more productive

firms. Greater competition increases both employment and output per match.

If n ≥ 1 firms approach a given worker, the firm with the highest productivity

hires the worker and the resulting match output is the maximum of n draws from

G(x). If no firms approach a given worker, he is unemployed and produces zero

output. Let HG(x|n) = G(x)n be the cdf of the worker’s output conditional on n

firms arriving. To obtain the unconditional cdf HG(x; θ), the conditional cdf HG(x|n)

is weighted by the Poisson probability that n firms approach:

(2) HG(x; θ) =∞∑n=0

θne−θ

n!G(x)n = e−θ(1−G(x)).

Aggregate Output. For any distribution G(x), aggregate output is given by Y =

fG(θ)L where fG(θ) is output per capita, y ≡ Y/L. Output per capita is just the

expected value of the endogenous distribution HG(x; θ) given by (2), namely

(3) fG(θ) =

∫ ∞1

θe−θ(1−G(x))xg(x)dx.

Clearly, the aggregate production function Y = fG(θ)L exhibits constant returns

to scale. For any well-behaved distribution G, the general function fG given by (3)

has some desirable features. It is twice differentiable, increasing and strictly concave,

and it satisfies a "weak" version of the Inada conditions. More precisely, it satisfies

all of the standard Inada conditions except that limθ→0 f′G(θ) is finite.14

Proposition 1. If G is well-behaved, the function fG has the following properties:

(i) f ′G(θ) > 0; (ii) f ′′G(θ) < 0; (iii) fG(0) = 0; (iv) limθ→∞ fG(θ) = +∞; (v)

limθ→∞ f′G(θ) = 0; and (vi) limθ→0 f

′G(θ) = EG(x) ≥ xmin.

Proof. See Appendix A1.

14The Inada condition limθ→0f ′(θ) = ∞ is a suffi cient but not a necessary condition for theexistence of a steady state equilibrium in most applications in macroeconomics. Generally, what isstrictly necessary is that limθ→0f ′(θ) is finite but suffi ciently large. Here, limθ→0f ′(θ) = EG(x) ≥xmin. We have normalized xmin = 1 for simplicity but in principle it could be any finite number.

8

Since these will prove useful later, I also provide f ′G(θ) and f ′′G(θ) here:

f ′G(θ) =

∫ ∞1

e−θ(1−G(x))(1−G(x))dx+ e−θ(4)

f ′′G(θ) = −(∫ ∞

1

e−θ(1−G(x))(1−G(x))2dx+ e−θ)

(5)

Considered as a property of the function fG(θ), the aggregate elasticity of substi-

tution between capital and labor is given by the following expression found in Arrow,

Chenery, Minhas, and Solow (1961).15 Since this elasticity is variable, we write σG(θ).

(6) σG(θ) =−f ′G(θ)(fG(θ)− θf ′G(θ))

θfG(θ)f ′′G(θ).

Proposition 2 establishes that for any well-behaved distribution G, the elasticity of

substitution σG(θ) is always less than or equal to one for any value of θ.16 Moreover,

the elasticity σG(θ) converges to one in the limit as θ →∞.

Proposition 2. If G is well-behaved, the aggregate elasticity of substitution σG(θ) is

always less than or equal to one. In the limit as θ →∞, we have σG(θ)→ 1.

Proof. See Appendix A2.

Since the elasticity of substitution σG(θ) is variable, Proposition 2 does not imply

that the aggregate production function converges asymptotically to a Cobb-Douglas

function for any well-behaved distribution G, but only that the value of the variable

elasticity of substitution converges to one as θ →∞. We focus on finite θ.

Example: Pareto Distribution

While all of the key results of this paper hold for any distribution G, our lead

example is the Pareto or power law distribution. This distribution is frequently

used in trade models to represent the productivity distribution across firms.17 Let15Note that this is different to the local elasticity of substitution between capital and labor, which

is zero since the local production technology for each match is essentially Leontief as it combinesone worker and one unit of capital.16In this setting, (6) is the natural definition. However, Proposition 2 also holds when the produc-

tion function and the elasticity of substitution are defined in a more conventional manner, so that yis a function of κ ≡ K/Le where Le is the number of employed workers. (See Appendix A3.)17Gabaix (2009) provides a review of the numerous applications and results regarding the Pareto

or power law distribution in economics and finance. See also Gabaix (2014).

9

G(x) = 1 − x−1/λ for x ∈ [1,∞) and G(x) = 0 otherwise. The single parameter

λ ∈ (0, 1) is often called the tail index since a higher value of λ implies fatter tails.

We refer to λ as the shape parameter.18

The Pareto distribution is well-behaved in the sense of Definition 1. In fact, it is

the unique distribution G such that εG(x) = 1/λ and ε′G(x) = 0.

If G is Pareto, the endogenous distribution H(x; θ) is

(7) H(x; θ) =

{e−θx

−1/λif x ∈ [1,∞)

e−θ otherwise

Notice that in the limit as θ → ∞, this distribution (appropriately normalized)converges asymptotically to a Type II Extreme Value Distribution or Fréchet distribu-

tion. Our focus is not on this extreme value distribution but on the exact distribution

for finite θ given by (7). The mass point at zero with probability mass u(θ) = e−θ is

crucial in a labor market setting as it corresponds to unemployment.

Before presenting the aggregate production function that arises when G is Pareto,

we introduce an important function that is a generalization of the Gamma function

Γ(s). Fact 1 summarizes some useful properties of this key function.19

Definition 2. For any s, x ∈ R+, the Lower Incomplete Gamma function is

(8) γ(s, x) ≡∫ x

0

ts−1e−t dt.

Fact 1. The function γ(s, x) has the following properties: (i) the recurrence relation:

γ(s, x) = (s− 1)γ(s− 1, x)− xs−1e−x; (ii) ∂∂xγ(s, x) = xs−1e−x > 0; (iii) ∂

∂sγ(s, x) =∫ x

0ts−1e−t(ln t)dt; (iv) limx→∞ γ(s, x) = Γ(s); and (v) γ(1, x) = 1− e−x.

For the Pareto distribution, output per capita f(θ) assumes the following form:

(9) f(θ) = θλγ(1− λ, θ),

To see this, let t = θ(1−G(x)) in (3) and then use Definition 1 to obtain (9).

18The mean of the Pareto distribution G(x) = 1−x−1/λ is 11−λ and the variance (defined only for

λ < 1/2) is λ2

(1−2λ)(1−λ)2 , both of which are increasing in λ.19For standard properties such as Fact 1, see Andrews, Askey, and Roy (2000) for example.

10

Using the fact that θ = K/L, the aggregate production function Y = f(θ)L is

(10) Y = γ(1− λ, θ)KλL1−λ.

Importantly, this function is not Cobb-Douglas since the term γ(1 − λ, θ) is not

constant but depends on θ = K/L. Consistent with Proposition 2, the aggregate

elasticity of substitution σG(θ) is always less than one for finite θ.

The fact that the production function is not generally Cobb-Douglas arises from

a crucial difference between this aggregation result and that found in Jones (2005).

Jones considers a large number of production units, each of which uses a local Leontief

production technology. He then takes the convex hull of available ideas to derive

a "global" production function, where ideas represent different ways of combining

capital and labor to produce output. Jones considers the convex hull across the

entire economy, and takes the limit as the number of ideas becomes large. Jones’

global production function is asymptotically Cobb-Douglas in the long run as the

total number of ideas across the economy grows over time.20

In contrast with Jones’approach, we take the highest productivity among the firms

competing for each individual worker and then aggregate across all workers. Since the

expected number of firms approaching a given worker is finite, we consider the exact

distribution which arises for finite θ, namely H(x; θ) with continuous support [1,∞)

plus a mass point at zero corresponding to unemployment. The resulting aggregate

production function is not generally Cobb-Douglas due to the existence of frictional

unemployment. This general frictional setting is the unique focus of the present paper.

While our focus is on the unified aggregate production and matching function

given by (9), we can isolate both the matching function and the limiting production

function as follows. Setting λ = 0 yields the special case where firms are homogeneous

with productivity normalized to one. Since γ(1, θ) = 1−e−θ, the aggregate productionfunction f(θ) collapses to the urn-ball matching function, m(θ). On the other hand,

if we consider the special limiting case where θ →∞ and unemployment u(θ)→ 0, all

workers are matched and the aggregate production function is asymptotically Cobb-

Douglas since limθ→∞f(θ)

Aθλ= 1, where A = Γ(1− λ) is a constant.21

20See also Caselli and Coleman (2006), who examine the world technology frontier; and Growiec(2008), who generalizes some of Jones’aggregation results.21An alternative way to obtain an exact Cobb-Douglas production function is to consider the limit

of the distribution H(x; θ) (appropriately normalized) as xmin → 0.

11

IV Equilibrium

Similarly to the large economy version of the competing auctions framework found

in Peters and Severinov (1997), the expected payoffs for workers and firms are just the

expected payoffs for sellers and buyers respectively in a second-price auction where

the number of buyers is a Poisson random variable with parameter θ.22

To start with, workers choose a single reservation wage w∗R equal to the value of

non-market activity z since it is optimal for sellers to set a reserve price equal to their

outside option.23 This result is derived explicitly in Albrecht, Gautier, and Vroman

(2012), which amends an earlier result in Peters and Severinov (1997). In Appendix

A4, we derive the result that w∗R = z holds in this particular setting.

The equilibrium ratio of firms to workers θ∗ is determined by a zero profit condition

that equates the expected payoff for entering firms and the cost of entry r. For any

distribution G, the equilibrium labor market tightness θ∗ solves the following:

(11)∫ ∞

1

e−θ(1−G(x))(1−G(x))dx+ e−θ(1− z) = r,

using the fact that w∗R = z. See Appendix A4 for a derivation.

The left-hand side of equation (11) differs slightly from the expected payoff for

buyers derived in Peters and Severinov (1997) because here z ∈ [0, 1] and G has

support [1,∞). Since we allow the possibility that z < xmin = 1, we relax the "no

gap" assumption found in the competing auctions literature.

Proposition 3 presents some comparative statics results. Expected output per

match is defined as pG(θ) ≡ fG(θ)/m(θ). Equilibrium output per capita is y∗ = fG(θ∗)

and equilibrium output per match is p∗ = pG(θ∗).

Proposition 3. If G is well-behaved, there exists a unique equilibrium θ∗ > 0. We

have: (i) θ∗ is decreasing in the value of non-market activity z and the cost of capital

r; (ii) unemployment u∗ is increasing in both z and r; (iii) output per capita y∗ is

decreasing in both z and r; (iv) output per match p∗ is decreasing in both z and r.

Proof. See Appendix A5.

22McAfee and McMillan (1987) introduced auctions with a stochastic number of bidders.23McAfee (1993) proved that in a general setting with private independent values and competing

sellers, the equilibrium outcome is that sellers post auctions with reserve price equal to their ownvaluations, i.e. wR = z in this setting.

12

These results are intuitive. A higher value of non-market activity z means that

firms are deterred by the lower expected profits in one-on-one meetings, so θ∗ is lower

and unemployment is higher. A lower cost of capital r implies greater firm entry, so

θ∗ is higher and unemployment is lower. In the limit as r → 0, we have θ∗ →∞ and

hence unemployment goes to zero.

Equilibrium output per capita y∗ is decreasing in both z and r. Since there is

only the indirect effect through θ∗, this result follows immediately from the fact that

f ′G(θ) > 0 and θ∗ is decreasing in both z and r. Importantly, output per capita fG(θ)

is increasing not only because the matching probability m(θ) is increasing in θ, but

also because expected output per match pG(θ) is increasing in θ. Equilibrium output

per match p∗ is therefore decreasing in both z and r.

Informational and Matching Rents. Competing auctions models such as Peters

and Severinov (1997) and Albrecht et al. (2014) implicitly assume that sellers’valu-

ations are always greater than or equal to the minimum buyers’valuation, i.e. there

is no "gap".24 The greater generality of the present setting is important in a labor

market context since it allows both "informational" and "matching" rents to coexist.

Informational rents arise due to the presence of private information and the fact

that firms’productivities ("valuations") are independent. Workers observe only the

distribution, not the actual realized values drawn by firms, hence workers cannot

extract the full surplus.25 The informational rents available to firms are reflected in

the first term on the left-hand side of (11).

Matching rents arise from the existence of a positive match surplus for the least

productive firm when xmin − z > 0. Even if a firm has the minimum productivity,

xmin = 1, it is still possible to earn a profit, 1 − z, if they are lucky enough to be

matched with a worker in a one-on-one meeting, which occurs with probability e−θ

for the firm. The matching rents available to firms are reflected in the second term on

the left-hand side of (11). The value of these matching rents disappears if z = xmin,

and the probability of obtaining these rents disappears in the limit as θ →∞.A positive "gap", xmin − z, is necessary in directed search models where all firms

24In Peters and Severinov (1997), sellers’valuation is zero and buyers’valuations are drawn froma distribution with support [0,1]. In Albrecht et al. (2014), both sellers’and buyers’valuations aredrawn from distributions with support [0,1].25Guerrieri (2008) introduces informational rents in a competitive search economy, while Kennan

(2010) incorporates informational rents in a DMP style search model.

13

have the same productivity, such as Julien et al. (2000), since only matching rents

exist. We can recover the zero profit condition in such a model, e−θ(1 − z) = r, in

the special case where the distribution G is degenerate. On the other hand, setting

xmin = z delivers the corresponding condition for a competing auctions environment

where only informational rents exist. In general, both informational and matching

rents exist simultaneously in the present setting. I discuss this further in Section V.

Example: Pareto Distribution

For the Pareto distribution, setting G = 1− x−1/λ in (11) and applying definition

(8), we obtain the following zero profit condition:

(12) λθλ−1γ(1− λ, θ) + e−θ(1− z) = r.

Proposition 4 presents some comparative statics results regarding the shape pa-

rameter λ. As we will see in Section V, the parameter λ can be seen as capturing the

degree of informational rents available to firms. Proposition 4 establishes that the

equilibrium labor market tightness θ∗ is increasing in λ and hence the equilibrium un-

employment rate u∗ is decreasing in λ. Equilibrium output per capita y∗ and output

per match p∗ are also increasing in λ.

Proposition 4. If G is Pareto with shape parameter λ, then: (i) θ∗ is increasing in

λ; (ii) unemployment u∗ is decreasing in λ; (iii) output per capita y∗ is increasing in

λ; and (iv) output per match p∗ is increasing in λ.

Proof. See Appendix A6.

V Distribution

For any distribution G, expected wages is wG(θ) = fG(θ)−rθ. This is the expectedpayoff from market activity for all workers, including the unemployed. Substituting

in (3) and (11), expected wages is given by

(13) wG(θ) =

∫ ∞1

θe−θ(1−G(x))

(x− 1−G(x)

g(x)

)g(x)dx− (1− z)θe−θ.

14

Again, this can be reconciled with the expected payoff for sellers in Peters and Sev-

erinov (1997) by noting that here z ∈ [0, 1] and G has support [1,∞), and also that

expected wages wG(θ) includes only the payoff from market activity.

While expected wages are determined by the labor market tightness θ, there is

residual wage dispersion among ex ante identical workers. This is because the actual

wage paid to a worker depends on both the number of firms competing to hire the

worker and the specific productivity draws of those firms. The wage distribution is

the distribution of the second highest productivity draw from n ≥ 2 firms where n is

Poisson-distributed with parameter θ, or the reservation wage wR = z if n = 1.

In general, the marginal product of labor is MPL = fG(θ)− θf ′G(θ) and the mar-

ginal product of capital is MPK = f ′G(θ). Factors are paid their marginal products

if and only if f ′G(θ) = r. Comparing the zero profit condition (11) and equation (4),

this occurs in two distinct cases: (i) if z = 0; and (ii) in the limiting case where

θ →∞. If z > 0, workers are paid more than their marginal product for finite θ.

Of course, the unified nature of the aggregate production function affects the

way in which the marginal products of labor and capital should be interpreted here.

The marginal product of labor represents the effect on aggregate output of an extra

potential worker, who may end up either employed or unemployed depending on

the matching outcomes. The marginal product of capital represents the marginal

contribution of an extra unit of hired capital, which may end up either utilized or

unutilized depending on whether or not the firm is successful.

Factor Income Shares. Imagine there are owners of capital who are paid the cost

of capital r by firms who enter. Since there is a zero profit condition for firms, aggre-

gate income is split between workers and the owners of capital. For any distribution

G, the share of income going to capital is sK(θ;G) = rθ/fG(θ) and labor’s share is

sL(θ;G) = 1− sK(θ;G). Using (11) and (3), we have

(14) sK(θ;G) =

∫∞1e−θ(1−G(x))(1−G(x))dx+ e−θ(1− z)∫∞

1e−θ(1−G(x))xg(x)dx

.

Proposition 5 characterizes the asymptotic behavior of factor shares in the limit

as θ →∞. This result is consistent with Gabaix et al. (2014) which establishes resultsregarding the limiting value of markups when the number of firms is large.26

26We can reconcile this result with Gabaix et al. (2014) as follows. In the limit as θ →∞, capital’s

15

Proposition 5. If G is well-behaved, in the limit as θ → ∞ we have sK(θ;G) →limx→∞ 1/εG(x) = α for some α ∈ [0, 1].

Proof. See Appendix A7.

In the limit as θ → ∞ and unemployment disappears, labor’s share approaches

its upper bound, sL(θ;G)→ 1− α. At the other extreme, in the limit as θ → 0 and

unemployment is pervasive, labor’s share approaches its lower bound, sL(θ;G) →z/EG(x). Even in the presence of extremely high unemployment, labor’s share does

not approach zero unless the value of non-market activity z is zero.

Our focus is on the behavior of factor shares outside these limiting cases. In

general, factor shares vary with the value of non-market activity z and the labor

market tightness ratio θ, in a way that depends on the underlying distribution G.

Lemma 1 states that labor’s share is increasing in θ. This result is intuitive since

θ is a measure of the degree of competition between firms to hire workers. Greater

competition between firms increases labor’s share of income.

Lemma 1. If G is well-behaved, labor’s share sL(θ;G) is increasing in θ.

Proof. See Appendix A8.

The equilibrium capital share s∗K is given by (14) where the labor market tightness

ratio θ∗ solves (11). Proposition 6 presents some general comparative statics results.

Proposition 6. If G is well-behaved, the equilibrium labor share s∗L is increasing in

the value of non-market activity z and decreasing in the cost of capital r.

Proof. See Appendix A9.

Capital’s share is increasing in the rental rate of capital r, since there is only the

indirect effect on s∗K through θ∗. If r increases, θ∗ decreases, and since sK(θ;G) is

decreasing in θ by Lemma 1, this increases the capital share.

If the value of non-market activity z increases, the direct effect is that equilibrium

capital share s∗K should decrease. However, the indirect effect is that an increase in

z leads to a lower level of firm entry, reducing θ∗, which has a positive effect on s∗Kby Lemma 1. It turns out that the direct effect always dominates: capital’s share is

decreasing in z. Equivalently, labor’s share is increasing in z.

income share sK(θ;G) as defined here is identical to µLPn /Mn in the limit as n → ∞, where µLPnand Mn are defined as in Gabaix et al. (2014).

16

Example: Pareto Distribution

For the Pareto distribution, expected wages is given by the following:

(15) w(θ) = (1− λ)θλγ(1− λ, θ)− (1− z)θe−θ.

Using (9) and Fact 1, we have MPK = f ′(θ) = λθλ−1γ(1 − λ, θ) + e−θ, and using

(9), we have MPL = f(θ) − θf ′(θ) = (1 − λ)θλγ(1 − λ, θ) − θe−θ. It is clear frominspecting (12) and (15) that factors are paid their marginal products only either

when z = 0 or in the limit as θ →∞.If G is Pareto, Proposition 5 implies that sK → λ in the limiting case where

θ →∞. Of greater interest, however, is the more general expression for capital’s sharefor finite values of θ. Before presenting this expression, we introduce the following

function. Lemma 2 summarizes some of its key properties.

Definition 3. For any s ∈ R+ and x ∈ R+\{0}, ε(s, x) is the elasticity of γ(s, x)

with respect to x,

(16) ε(s, x) ≡ xse−x

γ(s, x).

Lemma 2. The elasticity ε(s, x) has the following properties: (i) ∂∂sε(s, x) > 0; (ii)

∂∂xε(s, x) < 0; (iii) limx→0 ε(s, x) = s; and (iv) limx→∞ ε(s, x) = 0.

Proof. See Appendix A10.

To obtain capital’s share, substituting εG(x) = 1/λ and (9) into (14) yields

(17) sK = λ+ (1− z)ε(1− λ, θ).

For the Pareto distribution, constant factor shares can arise in two distinct cases.

In the limit as θ →∞, the elasticity ε(1−λ, θ) goes to zero and sK → λ. In this case,

factors are paid their marginal products and we have both a Cobb-Douglas aggregate

production function and constant factor shares. Alternatively, if z = xmin = 1, the

right term in (17) disappears so sK = λ even for finite θ. In this case, the aggregate

production function is not Cobb-Douglas but factor shares are nonetheless constant.

17

For the Pareto distribution, the shape parameter λ measures the size of the in-

formational rents available to firms.27 Informational rents are captured by the first

term in (17) and matching rents are captured by the second term. In general, if

z < xmin = 1 and λ 6= 0, firms are paid both informational rents and matching rents.

As θ → ∞, the probability of a one-on-one meeting goes to zero and hence match-ing rents disappear, but informational rents remain and sK → λ. Alternatively, if

z = xmin = 1, matching rents disappear and again we have sK → λ. In the special

case where λ = 0, there is no private information and hence no informational rents,

but matching rents are available for z < xmin = 1 and finite θ.

We can consider the effect on equilibrium factor shares of changes in the underlying

distribution G by allowing the parameter λ to vary. Since λ measures the degree of

informational rents, we might expect capital’s share to be increasing in this parameter.

On the other hand, a higher λ induces greater firm entry and more competition,

leading capital’s share to decrease. The net effect is ambiguous.

Proposition 7. If G is Pareto with shape parameter λ, the equilibrium labor share

s∗L is decreasing in λ if the value of non-market activity satisfies z > 1/(2− λ).

Proof. See Appendix A11.

Proposition 7 provides a condition on z and λ that is suffi cient (but not necessary)

for the equilibrium capital share to be increasing in the shape parameter λ.28 This

condition states, equivalently, that the minimum match surplus, xmin − z, cannot betoo high, i.e. 1− z < (1− λ)/(2− λ).

VI Effi ciency

For any well-behaved distribution G, the economy is constrained effi cient. Subject

to the constraints of the matching frictions, the socially optimal labor market tightness

ratio, θP , equals the decentralized equilibrium ratio θ∗. Equivalently, the equilibrium

unemployment rate is socially optimal. This result is common in models of directed

27This is because it represents the information content of a firm’s productivity draw. More pre-cisely, the information entropy of the Pareto distribution is lnλ + λ + 1 for λ 6= 0 so there is aone-to-one mapping between λ and the information entropy.28By Assumption 1, EG(x) > z + r. If G is Pareto, the suffi cient condition in Proposition 7 is

consistent with Assumption 1 if and only if r < 1(1−λ)(2−λ) . Since λ ∈ [0, 1], r < 1/2 will suffi ce.

18

or competitive search: constrained effi ciency generally does obtain in large markets,

at least in a static setting.29

The social planner’s objective is to maximize total output plus the value of non-

market activity minus the total costs of entry, i.e. to maximize Λ(θ) = fG(θ)+ze−θ−rθ. The social planner’s solution θP satisfies f ′G(θ)−ze−θ = r, and the equilibrium θ∗

satisfies (11). Substituting (4) into the first order condition, it is clear that θ∗ = θP .

Using (5), the second order condition, f ′′G(θ) + ze−θ < 0, also holds in this setting.

This extends the result regarding the constrained effi ciency of competing auctions

environments found in Albrecht et al. (2014) by allowing for both informational and

matching rents.30 In turn, it enables us to recover the constrained effi ciency of directed

search models such as Julien et al. (2000) as the special case where the distribution

G is degenerate. In such models, only matching rents exist.

Generalized Hosios Condition. The well-known Hosios condition introduced in

Hosios (1990) states that constrained effi ciency obtains when workers’ bargaining

power equals the elasticity of the matching function with respect to unemployed

workers. To understand the relationship between the present model and some bench-

mark environments, it is useful to generalize this condition to allow for settings where

output per match is endogenous.

Consider a general environment in which output per capita f(θ) is broken into

two components: expected output per match, p(θ), and the matching probability for

workers, m(θ). That is, let f(θ) = p(θ)m(θ). Let Λ(θ) denote the social surplus per

worker, given by Λ(θ) = f(θ) + z(1 − m(θ)) − rθ. The first order condition for thesocial planner’s problem is given by

(18)dΛ(θ)

dθ= p′(θ)m(θ) + p(θ)m′(θ)− zm′(θ)− r = 0.

Suppose also that the second order condition holds, i.e. f ′′(θ)− zm′′(θ) < 0.

Let εy,θ be the elasticity of f(θ) with respect to θ, and let η(θ) be the elasticity

of the matching function m(θ) with respect to θ. Constrained effi ciency holds in this

29Guerrieri (2008) shows that a competitive search economy with private information can beconstrained ineffi cient in a dynamic setting where workers’outside option is endogenous.30Of course, Albrecht et al. (2014) allow for seller heterogeneity, as well as both ex post and ex

ante buyer heterogeneity, so the present setting is less general in that regard.

19

general environment if and only if the decentralized equilibrium θ∗ satisfies

(19) εy,θ =rθ

f(θ)+zη(θ)

p(θ).

This generalizes the standard Hosios condition to environments with both matching

frictions and endogenous output per match. If the value of non-market activity z is

zero, this condition can be stated simply: constrained effi ciency obtains when labor’s

share of income equals its elasticity in the aggregate production function.

Examples. As the first benchmark, consider a standard competitive economy where

output per worker is f(θ) and there is no unemployment. Clearly, condition (19) holds

since sK = rθ/f(θ), which equals the output elasticity εy,θ since f ′(θ) = r.

Now consider a DMP-style environment where wages are determined by general-

ized Nash bargaining, there is an exogenous matching function m(θ), and exogenous

output per match p(θ) = p. Workers’bargaining parameter is β, r is the flow vacancy

cost, and z is the value of non-market activity. It is well known that the decentralized

equilibrium θ∗ solves

(20)m(θ)

θ(1− β)(p− z) = r,

Substituting f(θ) = pm(θ) into (19) and using (20), we recover the familiar Hosios

condition: constrained effi ciency holds if and only if η(θ) = 1− β.For the sake of comparison, we can also consider a setting where output per

match is endogenous, p(θ) = Aθα, but wages are determined by Nash bargaining.

Substituting (20) into (19) with p(θ) instead of p, we have effi ciency if and only if

(21) α = (1− β)

(p(θ)− zp(θ)

)+zη(θ)

p(θ).

Clearly, there is no reason why constrained effi ciency would hold since both the match-

ing elasticity η(θ) and the parameter α governing the production technology are in-

dependent of the bargaining parameter β.

In the present model, we have a unified aggregate production and matching func-

tion, fG(θ). Using (3), (4), and (14), it is straightforward to verify condition (19) for

any well-behaved distribution G of firm productivities.

20

For the Pareto example, we have f(θ) = θλγ(1−λ, θ) and εy,θ = λ+ ε(1−λ, θ). Ifz = 0, we have εy,θ = sK by (17) and hence (19) clearly holds. In general, zη(θ)/p(θ) =

zε(1 − λ, θ), so the right hand side of (19) is also λ + ε(1 − λ, θ) and (19) holds.

Importantly, the same parameter governs both production and distribution. While

this is particularly clear in the limiting case where θ →∞ and we have f(θ)→ Aθλ

and sK = λ, it is also true more generally when unemployment is present.

VII Conclusion

The standard neoclassical approach tightly links production and distribution.

Diamond—Mortensen-Pissarides style search models introduce labor market frictions

and unemployment but break this connection: distribution is governed by the bar-

gaining parameter and constrained effi ciency does not hold. Directed or competitive

search models provide a way to achieve constrained effi ciency, but the mapping back

into the standard neoclassical paradigm is unclear.

This paper offers a way forward by developing a unified approach to production,

matching, and distribution. The theory is simple and parsimonious and it delivers

both constrained effi ciency and a tractable aggregate production function. For any

well-behaved distribution of firm productivities, this function exhibits standard neo-

classical features and an elasticity of substitution between capital and labor that is

always less than or equal to one. In general, factor income shares vary with the degree

of competition between firms to hire workers, the value of non-market activity, and

the nature of the underlying firm productivity distribution.

There is a great deal of further research to be done. The theory developed here

maps out the key outlines of a novel approach, but it is perhaps too simple and

parsimonious as it stands: various extensions and modifications may be required

before its predictions can be tested empirically. In Mangin (2014), I extend the model

to a dynamic setting and examine its predictions regarding the behavior of factor

income shares. Possible directions for future theoretical work include: introducing an

intensive margin for capital, incorporating on-the-job search or multiple applications,

and allowing for ex ante worker and firm heterogeneity.

21

Appendix A —Proofs

A1. Proof of Proposition 1. Let fG(θ) =

∫ ∞1

θe−θ(1−G(x))xg(x)dx. Applying

Leibniz’s rule, we have

(22) f ′G(θ) =

∫ ∞1

xg(x)e−θ(1−G(x))dx−∫ ∞

1

θxg(x)e−θ(1−G(x))(1−G(x))dx.

By integration by parts on the right integral, using the fact that limx→∞ x(1−G(x)) = 0,which follows from the finite mean assumption, we have(23)∫ ∞

1

θxg(x)e−θ(1−G(x))(1−G(x))dx = −e−θ −∫ ∞

1

e−θ(1−G(x))((1−G(x))− xg(x))dx,

Substituting (23) into (22),

(24) f ′G(θ) =

∫ ∞1

e−θ(1−G(x))(1−G(x))dx+ e−θ > 0,

and part (i) is proved. Next, we use Leibniz’rule again to prove part (ii),

(25) f ′′G(θ) = −(∫ ∞

1

e−θ(1−G(x))(1−G(x))2dx+ e−θ)< 0.

It is clear that f(0) = 0 and limθ→∞ f′G(θ) = 0, so parts (iii) and (v) hold. Now con-

sider limθ→∞ fG(θ). Changing variables by setting t = 1 − G(x), we have fG(θ) =

θ

∫ 1

0

e−θtG−1(1 − t)dt. Defining G−1(y) = 0 for y < 0, we have G−1(1 − t) = 0 for

t > 1 so we can write fG(θ) = θ

∫ ∞0

e−θtG−1(1 − t)dt. We can now apply the ini-

tial value theorem for Laplace transforms, which states that for any piecewise continu-ous function φ(t), limθ→∞ θ

∫∞0e−θtφ(t)dt = limt0→0 φ(t0). So we have limθ→∞ fG(θ) =

limt0→0G−1(1 − t0) = G−1(0) = +∞, and part (iv) holds. Finally, using (22), we have

limθ→0 f′G(θ) = limθ→0

∫∞1xg(x)e−θ(1−G(x))dx =

∫∞1xg(x)dx = EG(x), so (vi) holds.

A2. Proof of Proposition 2. The following lemma will turn out to be useful.

Lemma 3. Let α(.), β(.) and ϕ(.) be positive functions defined on [1,∞). Suppose thatα′(x) ≤ 0 and β′(x) < 0. Then

∫∞1α(x)h(x)dx ≤

∫∞1α(x)h(x)dx, where h(x) ≡

ϕ(x)∫∞1 ϕ(x)dx

and h(x) ≡ ϕ(x)β(x)∫∞1 ϕ(x)β(x)dx

.

22

Proof. Since β′(x) < 0, β−1 exists and h(x)− h(x) ≥ 0 holds if and only if

x ≤ xc ≡ β−1

(∫∞1ϕ(x)β(x)dx∫∞1ϕ(x)dx

),

for some critical value xc ∈ [1,∞). Now for any x ∈ [1, xc], h(x) − h(x) ≥ 0 andα(x) ≥ α(xc) since α′(x) ≤ 0, so

(26)∫ xc

1

α(x)(h(x)− h(x))dx ≥∫ xc

1

α(xc)(h(x)− h(x))dx.

For any x ∈ [xc,∞), α(x) ≤ α(xc) since α′(x) ≤ 0, but here h(x)− h(x) ≤ 0, so

(27)∫ ∞xc

α(x)(h(x)− h(x))dx ≥∫ ∞xc

α(xc)(h(x)− h(x))dx.

Using inequalities (26) and (27), we have∫ ∞1

α(x)(h(x)− h(x))dx

=

∫ xc

1

α(x)(h(x)− h(x))dx+

∫ ∞xc

α(x)(h(x)− h(x))dx

≥∫ xc

1

α(xc)(h(x)− h(x))dx+

∫ ∞xc

α(xc)(h(x)− h(x))dx

= α(xc)

(∫ ∞1

h(x)dx−∫ ∞

1

h(x)dx

)= 0.

Using Lemma 3, we first prove that if G is well-behaved then σG(θ) ≤ 1. Starting withthe definition found in Arrow et al. (1961) (p. 229), we have

(28) σG(θ) =−f ′G(θ)(fG(θ)− θf ′G(θ))

θfG(θ)f ′′G(θ).

Let G(x) = 1−G(x). Inserting f ′G(θ) from (24) and f ′′G(θ) from (25) into (28), we have

σG(θ) =

(∫∞1e−θG(x)G(x)dx+ e−θ

) (∫∞1e−θG(x)xg(x)dx−

(∫∞1e−θG(x)G(x)dx+ e−θ

))(∫∞1θe−θG(x)xg(x)dx

) (∫∞1e−θG(x)G(x)2dx+ e−θ

) .

Using (23) and simplifying further, we have

(29) σG(θ) =

(∫∞1e−θG(x)G(x)dx+ e−θ

) (∫∞1e−θG(x)xg(x)G(x)dx

)(∫∞1e−θG(x)xg(x)dx

) (∫∞1e−θG(x)G(x)2dx+ e−θ

) .

23

Multiplying out (29) yields

σG(θ) =

(∫∞1e−θG(x)G(x)dx

) (∫∞1e−θG(x)xg(x)G(x)dx

)+ e−θ

(∫∞1e−θG(x)xg(x)G(x)dx

)(∫∞1e−θG(x)xg(x)dx

) (∫∞1e−θG(x)G(x)2dx

)+ e−θ

(∫∞1e−θG(x)xg(x)dx

) .

Now since G(x) ≤ 1 and both integrands are positive,∫∞

1e−θG(x)xg(x)G(x)dx ≤

∫∞1e−θG(x)xg(x)dx.

In order to show that σG(θ) ≤ 1, it is suffi cient to show that

(30)

∫∞1e−θG(x)G(x)dx

∫∞1e−θG(x)xg(x)G(x)dx∫∞

1e−θG(x)xg(x)dx

∫∞1e−θG(x)G(x)2dx

≤ 1.

Rearranging, and using the definition of εG(x), this inequality is equivalent to

(31)

∫∞1

(1/εG(x))e−θG(x)xg(x)dx∫∞1e−θG(x)xg(x)dx

≤∫∞

1(1/εG(x))e−θG(x)xg(x)G(x)dx∫∞

1e−θG(x)xg(x)G(x)dx

.

We can now apply Lemma 3, where α(x) = 1/εG(x), ϕ(x) = e−θG(x)xg(x), and β(x) =G(x). We have α(x) ≥ 0, ϕ(x) ≥ 0 and β(x) ≥ 0. By assumption, ε′G(x) ≥ 0, soα′(x) ≤ 0 and β′(x) = −g(x) < 0. By Lemma 3, we have∫∞

1(1/εG(x))ϕ(x)dx∫∞

1ϕ(x)dx

≤∫∞

1(1/εG(x))ϕ(x)β(x)dx∫∞

1ϕ(x)β(x)dx

.

We now prove that σG(θ) converges to one in the limit as θ →∞. Starting with (29)and letting t = 1−G(x), so x = G−1(1− t) for t ∈ (0, 1],

σG(θ) =

(∫ 1

0e−θt

(t

g(G−1(1−t))

)dt+ e−θ

)(∫ 1

0e−θttG−1(1− t)dt

)(∫ 1

0e−θtG−1(1− t)dt

)(∫ 1

0e−θt

(t2

g(G−1(1−t))

)dt+ e−θ

)=

(∫ 1

0e−θt

(G−1(1− t) t

g(G−1(1−t))G−1(1−t)

)dt+ e−θ

)(∫ 1

0e−θttG−1(1− t)dt

)(∫ 1

0e−θtG−1(1− t)dt

)(∫ 1

0e−θt

(tG−1(1− t) t

g(G−1(1−t))G−1(1−t)

)dt+ e−θ

) .Now define f1(t) = G−1(1 − t) and f2(t) = t

g(G−1(1−t))G−1(1−t) for t ∈ (0, 1], and let

f1(t) = f2(t) = 0 for t > 1. Then limθ→∞ σG(θ) is given by

(32) limθ→∞

σG(θ) = limθ→∞

(∫∞0e−θtf1(t)f2(t)dt+ e−θ

) (∫∞0e−θttf1(t))dt

)(∫∞0e−θtf1(t)dt

) (∫∞0e−θttf1(t)f2(t)dt+ e−θ

) .Let t0 ∈ (0, 1]. Multiplying each integral in (32) by θ and dividing both the numerator and

24

denominator by f1(t0)f2(t0) and t0f1(t0), we have

limθ→∞

σG(θ) = limθ→∞

(θ∫∞

0e−θt f1(t)f2(t)

f1(t0)f2(t0)dt+ θe−θ

f1(t0)f2(t0)

)(θ∫∞

0e−θt tf1(t)

t0f1(t0)dt)

(θ∫∞

0e−θt f1(t)

f1(t0)dt)(

θ∫∞

0e−θt tf1(t)f2(t)

t0f1(t0)f2(t0)dt+ θe−θ

t0f1(t0)f2(t0)

) .Using limit operations and applying the initial value theorem for Laplace transforms,

limθ→∞

σG(θ) =

(limt0→0

f1(t0)f2(t0)f1(t0)f2(t0)

+ 0)(

limt0→0t0f1(t0)t0f1(t0)

)(

limt0→0f1(t0)f1(t0)

)(limt0→0

t0f1(t0)f2(t0)t0f1(t0)f2(t0)

+ 0) = 1.

A3. Alternative version of Proposition 2. The elasticity of substitutionσG(θ) is defined as a property of the function fG(θ), where θ = K/L and L is the la-bor force, or total number of potential workers. We can also consider the elasticity ofsubstitution σG(θ) defined as a property of the function fG(κ) where κ ≡ K/Le and Le≡ (1 − e−θ)L, the number of employed workers. For simplicity, we write σ(θ) and f(θ)instead of σG(θ) and fG(θ) respectively.

We have κ = θ/(1 − e−θ). Since κ′(θ) = (1 − e−θ − θe−θ)/(1 − e−θ)2 > 0, κ(θ)is invertible and we can write θ(κ). Let g(κ) ≡ f(θ(κ)). The elasticity of substitutionbetween capital and employed workers, σ(θ), is given by

σ(θ) =−g′(κ)(g(κ)− κg′(κ))

κg(κ)g′′(κ).

In the limit as θ → ∞, it is clear that κ/θ → 1, so we have σ(θ) → σ(θ) and henceσ(θ) → 1 as θ → ∞. To show that σ(θ) ≤ 1 in general, it suffi ces to show thatσ(θ) ≤ σ(θ) for any θ since we have already proven that σ(θ) ≤ 1. Now g′(κ) = f ′(θ)θ′(κ)and g′′(κ) = f ′′(θ)(θ′(κ))2 + f ′(θ)θ′′(κ). So

σ(θ) =−f ′(θ)θ′(κ)(f(θ)− κf ′(θ)θ′(κ))

κf(θ)(f ′′(θ)(θ′(κ))2 + f ′(θ)θ′′(κ)).

To show that σ(θ) ≤ σ(θ), we need to prove that

σ(θ) =−f ′(θ)θ′(κ)(f(θ)− κf ′(θ)θ′(κ))

κf(θ)(f ′′(θ)(θ′(κ))2 + f ′(θ)θ′′(κ))≤ −f

′(θ)(f(θ)− θf ′(θ))θf(θ)f ′′(θ)

= σ(θ).

Substituting in κ = θ/(1 − e−θ) and using the fact that f(θ) ≥ 0 and f ′(θ) > 0 fromProposition 1, this is equivalent to showing that

(33)θ′(κ)((1− e−θ)f(θ)− θf ′(θ)θ′(κ))

(f ′′(θ)(θ′(κ))2 + f ′(θ)θ′′(κ))≥ f(θ)− θf ′(θ)

f ′′(θ).

25

Now κ′(θ) = (1− e−θ − θe−θ)/(1− e−θ)2, so we have

θ′(κ) =(1− e−θ)2

1− e−θ − θe−θ .

Differentiating with respect to κ and simplifying,

θ′′(κ) =e−θ(1− e−θ)(2− θ − e−θ(2 + θ))

(1− e−θ − θe−θ)2.

Rearranging (33), using the fact that f ′′(θ) < 0 from Proposition 1, we need to prove that

(34)(

1 +B − AB

)f(θ)− θf ′(θ) ≥ 0.

where

A =1− e−θθ′(κ)

and B =f ′(θ)θ′′(κ)

f ′′(θ)(θ′(κ))2.

Since f(θ)− θf ′(θ) ≥ 0, it suffi ces to show that 1+B−AB≥ 1, which is true if A ≤ 1. Since

1− e−θ ≤ θ′(κ), we have A ≤ 1 and hence (34) holds. So σ(θ) ≤ σ(θ) and σ(θ) ≤ 1.

A4. Derivation of zero profit condition. Let workers’reservation wage bewR. Suppose a firm draws x from the distribution G and n firms compete to hire the sameworker. If n = 1, the firm’s expected net payoff is just π1 =

∫∞1

(x − wR) dG(x) − r. Ifn ≥ 2, the expected net payoff is

(35) π2(x, n) = β(x, n)(x− w(x, n))− r,

where β(x, n) is the probability the firm is successful in hiring the worker and w(x, n) =E(Y n

2 |Y n1 = x) where Y n

2 is the second highest from n draws, and Y n1 is the highest.

Let H(y, n) be the distribution of Y n1 , i.e. H(y, n) = G(y)n. Now E(Y n

2 |Y n1 = x) =

E(Y n−11 |Y n−1

1 < x), so expected wages as a function of the highest productivity x and thenumber of firms n is w(x, n) = 1

H(x,n−1)

∫ x1y dH(y, n−1). Substituting w(x, n) into (35),

(36) π2(x, n) = β(x, n)

(x− 1

H(x,n−1)

∫ x

1

y dH(y, n− 1)

)− r.

Now β(x, n) is just G(x)n−1 = H(x, n− 1). Substituting into (36) and using integrationby parts, we obtain π2(x, n) =

∫ x1H(y, n − 1)dy − r. When n ≥ 2, the expected payoff

for a firm is π2(n) =∫∞

1π2(x, n)g(x)dx. Again integrating by parts, we obtain

(37) π2(n) = [π2(x, n)G(x)]∞1 −∫ ∞

1

d

dx[π2(x, n)]G(x)dx− r.

26

Now, ddx

[π2(x, n)] = ddx

(∫ x1H(y, n− 1)dy − r

)= H(x, n− 1). Also, [π2(x, n)G(x)]∞1 =

limx→∞ π2(x, n), which is∫∞

1H(y, n− 1)dy, since G(x)→ 1 as x→∞ and G(1) = 0.

Rearranging, we have

(38) π2(n) =

∫ ∞1

β(x, n)(1−G(x))dx− r.

The number of firms n approaching a given worker is a Poisson random variable withparameter θ, so the expected net payoff given n ≥ 2 is

(39) π2(θ) =

∫ ∞1

β(x)(1−G(x))dx− r,

where β(x) is the probability of being successful given that n ≥ 2, namely

(40) β(x) =1

1− e−θ∞∑n=2

e−θθn−1

(n− 1)!G(x)n−1 =

e−θ(1−G(x)) − e−θ1− e−θ .

Using π2(θ) and π1 and rearranging, the expected net payoff for a firm is

π(θ) =

∫ ∞1

e−θ(1−G(x))(1−G(x))dx+e−θ((∫ ∞

1

x g(x)dx−∫ ∞

1

(1−G(x))dx

)− wR

)−r.

By integration by parts and the fact that limx→∞ x(1−G(x)) = 0, we obtain

(41) π(θ) =

∫ ∞1

e−θ(1−G(x))(1−G(x))dx+ e−θ (1− wR)− r = 0.

Now suppose that workers choose wR in order to maximize the expected payoff fromboth market and non-market activity:

(42) w∗R = arg max f(θ∗(wR))− rθ∗(wR) + ze−θ∗(wR)

The first order condition holds if and only if f ′(θ)− r = ze−θ. Using (4) and (41), θ∗(wR)satisfies f ′(θ)− r = wRe

−θ and hence w∗R = z. Using w∗R = z , we obtain (11).

A5. Proof of Proposition 3. The zero profit condition (11) holds if and only

if F (θ) = 0 where F (θ) =

∫ ∞1

e−θ(1−G(x))(1 − G(x))dx + e−θ(1 − z) − r. Now F (θ) is

continuous in θ on [0, ∞) and F (θ) → −r as θ → ∞. If F (0) > 0, the intermediatevalue theorem implies there exists a θ > 0 such that F (θ) = 0. Using integration by parts,

F (0) =

∫ ∞1

(1−G(x))dx+(1−z)−r = EG(x)−z−r. So there exists a θ > 0 such that

π(θ) = 0 if Assumption 1 holds. Otherwise, no firms enter and θ = 0. To prove uniqueness

27

of the equilibrium θ∗, it suffi ces to show that F ′(θ) < 0. Applying Leibniz’integral rule,

(43) F ′(θ) = −(∫ ∞

1

e−θ(1−G(x)) (1−G(x))2 dx+ e−θ(1− z)

)< 0.

By the implicit function theorem, dθ∗

dz= −∂F/∂z

∂F/∂θ. Since z ≤ 1, we have

(44)dθ∗

dz=

−e−θ(∫∞1e−θ(1−G(x))(1−G(x))2dx+ e−θ (1− z)

) < 0.

Also, dθ∗

dr= −∂F/∂r

∂F/∂θwhere ∂F

∂r= −1. Using ∂F

∂θfrom above, we have

(45)dθ∗

dr=

−1(∫∞1e−θ(1−G(x))(1−G(x))2dx+ e−θ (1− z)

) < 0.

Finally, we prove that p′G(θ) > 0 where pG(θ) ≡ fG(θ)/m(θ). Differentiating, we havep′G(θ) > 0 if and only if h(θ) > 0 where

h(θ) ≡ f ′G(θ)θ

fG(θ)− m′(θ)θ

m(θ).

Since h(0) = 0, it suffi ces to prove h′(θ) > 0. Differentiating and simplifying, h′(θ) > 0 iff

(46)f ′′(θ)θf(θ) + f ′(θ)f(θ)− (f ′(θ))2θ

f(θ)2>e−θ(1− e−θ − θ)

(1− e−θ)2.

Using (28) and the result that σG(θ) ≤ 1, the left-hand side of (46) is greater than or equalto zero. So it suffi ces to show that 1 − e−θ − θ < 0, which is easily verified for all θ > 0.Hence p′G(θ) > 0. Now, since f ′G(θ) > 0, p′G(θ) > 0, and u′(θ) < 0, the fact that dθ∗

dz< 0

and dθ∗

dr< 0 implies that dy∗

dz< 0, dy

∗

dr< 0, dp

∗

dz< 0, dp

∗

dr< 0,du

∗

dz> 0, and du∗

dr> 0.

A6. Proof of Proposition 4. By the implicit function theorem, dθ∗

dλ= −∂F/∂λ

∂F/∂θ.

Using (43), we have

(47)∂F

∂θ= −

(λθλ−2γ(2− λ, θ) + e−θ(1− z)

).

Applying Fact 1 (iii) and simplifying yields

∂F

∂λ= θλ−1

(γ(1− λ, θ) + λ

∫ θ

0

t−λe−t(ln θ − ln t)dt

).

28

Again using (47), plus the fact that∫ θ

0t−λe−t(ln θ − ln t)dt > 0, we have

(48)dθ∗

dλ=θλ−1

(γ(1− λ, θ) + λ

∫ θ0t−λe−t(ln θ − ln t)dt

)λθλ−2γ(2− λ, θ) + e−θ(1− z)

> 0.

Next, let y∗ = f(θ∗(λ), λ) = θλγ(1− λ, θ). Then dy∗

dλ= ∂f

∂θdθ∗

dλ+ ∂f

∂λ. Now f ′(θ) > 0 and

dθ∗

dλ> 0 so it suffi ces to show that ∂f

∂λ> 0. Using Fact 1 (iii), we obtain

(49)∂f

∂λ= θλ

(∫ θ

0

t−λe−t(ln θ − ln t)dt

)> 0.

Similarly, dp∗

dλ= ∂p

∂θdθ∗

dλ+ ∂p

∂λ. Since p′(θ) > 0 and dθ∗

dλ> 0, it suffi ces to show that ∂p

∂λ> 0,

which follows from (49). Finally, the fact that du∗

dλ< 0 follows from dθ∗

dλ> 0 and u′(θ) < 0.

A7. Proof of Proposition 5. Starting with (14) and letting t = 1−G(x),

sK(θ;G) =

∫ 1

0e−θt

(t

g(G−1(1−t))

)dt+ e−θ(1− z)∫ 1

0e−θtG−1(1− t) dt

.

Now define f1(t) = G−1(1 − t) and f2(t) = tg(G−1(1−t))G−1(1−t) for t ∈ (0, 1], and let

f1(t) = f2(t) = 0 for t > 1. Rewriting, we have

sK(θ;G) =θ∫∞

0e−θtf1(t)f2(t)dt+ θe−θ(1− z)

θ∫∞

0e−θtf1(t)dt

.

Dividing both the numerator and denominator by f1(t0) for some t0 ∈ (0, 1], we have

limθ→∞

sK(θ;G) = limθ→∞

θ ∫∞0 e−θt f1(t)f2(t)f1(t0)

dt+ θe−θ(1−z)f1(t0)

θ∫∞

0e−θt f1(t)

f1(t0)dt

.

Using limit operations and the initial value theorem for Laplace transforms,

limθ→∞

sK(θ;G) =limt0→0

f1(t0)f2(t0)f1(t0)

+ 0

limt0→0f1(t0)f1(t0)

= limt0→0

f2(t0).

Substituting in f2(t) = tg(G−1(1−t))G−1(1−t) and t = 1−G(x), we have

limθ→∞

sK(θ;G) = limx→∞

1−G(x)

xg(x)= lim

x→∞1/εG(x).

29

By assumption, ε′G(x) ≥ 0 so 1/εG(x) is weakly decreasing in x. Also, for all x ≥ 0 wehave 1/εG(x) ≥ 0. Hence limθ→∞ sK(θ;G) = limx→∞ 1/εG(x) = α for some α ≥ 0.Finally, α ≤ 1 since 1/εG(x) ≤ 1 if and only if d

dxx(1 − G(x)) ≤ 0, and we know that

limx→∞ x(1−G(x)) = 0 and x(1−G(x)) ≥ 0 for all x.

A8. Proof of Lemma 1. Differentiating (14) with respect to θ, we have(50)

d

dθsK(θ;G) =

(−(∫∞

1e−θG(x)G(x)2dx+ e−θ(1− z)

) (∫∞1e−θG(x)xg(x)dx

)+(∫∞

1e−θG(x)G(x)xg(x)dx

) (∫∞1e−θG(x)G(x)dx+ e−θ(1− z)

) )(∫∞1e−θG(x)xg(x)dx

)2 .

Now ddθsK(θ;G) < 0 if and only if(∫ ∞

1

e−θG(x)G(x)xg(x)dx

)(∫ ∞1

e−θG(x)G(x)dx

)+ e−θ(1− z)

(∫ ∞1

e−θG(x)G(x)xg(x)dx

)<

(∫ ∞1

e−θG(x)G(x)2dx

)(∫ ∞1

e−θG(x)xg(x)dx

)+ e−θ(1− z)

(∫ ∞1

e−θG(x)xg(x)dx

).

Since G(x) < 1 for all x > 1, it suffi ces to show that(∫ ∞1

e−θG(x)G(x)xg(x)dx

)(∫ ∞1

e−θG(x)G(x)dx

)≤(∫ ∞

1

e−θG(x)G(x)2dx

)(∫ ∞1

e−θG(x)xg(x)dx

),

which is equivalent to inequality (30) derived in Appendix A2.

A9. Proof of Proposition 6. Starting with (14) and using (50) and (44),

ds∗Kdz

=∂sK∂θ

dθ∗

dz+∂sK∂z

=

−e−θ(−(∫∞

1e−θG(x)G(x)2dx+ e−θ(1− z)

) (∫∞1e−θG(x)xg(x)dx

)+(∫∞

1e−θG(x)G(x)xg(x)dx

) (∫∞1e−θG(x)G(x)dx+ e−θ(1− z)

) )(∫∞1e−θG(x)G(x)2dx+ e−θ(1− z)

) (∫∞1e−θG(x)xg(x)dx

)2

+−e−θ∫∞

1e−θG(x)xg(x) dx

.

Simplifying and rearranging,ds∗Kdz

< 0 if and only if(∫∞1e−θG(x)G(x)xg(x)dx

) (∫∞1e−θG(x)G(x)dx+ e−θ(1− z)

)(∫∞1e−θG(x)G(x)2dx+ e−θ(1− z)

) (∫∞1e−θG(x)xg(x)dx

) > 0,

which is true since both the numerator and denominator are positive.

30

A10. Proof of Lemma 2. Differentiating (16) with respect to s, we obtain

∂

∂sε(s, x) = xse−x

(∫ x0

(lnx− ln t)ts−1e−tdt

γ(s, x)2

)> 0.

Differentiating (16) with respect to x, we have

(51)∂

∂xε(s, x) =

xs−1e−x

γ(s, x)

(s− x− xse−x

γ(s, x)

)< 0.

To see this, observe that ∂∂xε(s, x) < 0 if and only if s− x < ε(s, x). Applying Fact 1 (i),

this is true provided that x > γ(s + 1, x)/γ(s, x). Multiplying both sides by xse−x andrearranging, this is true if and only if ε(s+1, x) > ε(s, x), which follows from ∂

∂sε(s, x) > 0.

Parts (iii) and (iv) follow from L’Hôpital’s rule.

A11. Proof of Proposition 7. We have s∗K = λ + (1 − z)ε(1 − λ, θ) whereθ∗(λ) solves the zero profit condition λθλ−1γ(1− λ, θ) + (1− z)e−θ = r. Rearranging thezero profit condition and substituting into the expression for capital share using Definition3, we obtain s∗K = rθ1−λ

γ(1−λ,θ) . Differentiating s∗K with respect to λ, we have

(52)ds∗Kdλ

= r∂

∂θ

(θ1−λ

γ(1− λ, θ)

)∂θ∗

∂λ+ r

∂

∂λ

(θ1−λ

γ(1− λ, θ)

).

Using Fact 1 (ii), we have

(53)∂

∂θ

(θ1−λ

γ(1− λ, θ)

)=

(1− λ)θ−λ

γ(1− λ, θ) −θ1−2λe−θ

γ(1− λ, θ)2.

Applying Fact 1 (iii) and simplifying,

(54)∂

∂λ

(θ1−λ

γ(1− λ, θ)

)=

−θ1−λ

γ(1− λ, θ)2

(∫ θ

0

t−λe−t(ln θ − ln t) dt

).

Letting B =∫ θ

0t−λe−t(ln θ − ln t)dt and then substituting (53) and (54) into (52),

(55)ds∗Kdλ

=rθ−λ

γ(1− λ, θ)

((1− λ− ε(1− λ, θ)) ∂θ

∗

∂λ− θB

γ(1− λ, θ)

)Applying Fact 1 (i), we have

ds∗Kdλ

> 0 if and only if ∂θ∗

∂λ> θB

γ(2−λ,θ) . Substituting in∂θ∗

∂λfrom

(48) and simplifying,ds∗Kdλ

> 0 if and only if

(56) γ(2− λ, θ)γ(1− λ, θ) > B(1− z)θ2−λe−θ.

31

Now suppose that z > 1/(2− λ), so 1− z < 1−λ2−λ . To prove (56), it suffi ces to show

(57) γ(2− λ, θ)γ(1− λ, θ) > B

(1− λ2− λ

)θ2−λe−θ.

To prove this, we introduce a generalized hypergeometric function defined by

F2,2(a1, a2; b1, b2; z) ≡∞∑n=0

(a1)n(a2)n(b1)n(b2)n

zn

n!,

where (a)n ≡ Γ(a+n)Γ(a)

, the Pochhammer symbol or ascending factorial function. Calculatingthe integral B, we have:

B = (ln θ)γ(1−λ, θ)−[(lnx)γ(1− λ, x)− x1−λ

(1− λ)2F2,2(1− λ, 1− λ; 2− λ, 2− λ;−x)

]θ0

.

As limx→0x1−λ

(1−λ)2F2,2(1− λ, 1− λ; 2− λ, 2− λ;−x) = limx→0(lnx)γ(1− λ, x) = 0,

(58) B =θ1−λ

(1− λ)2F2,2(1− λ, 1− λ; 2− λ, 2− λ;−θ).

Inequality (57) can now be stated purely in terms of generalized hypergeometric functionsusing γ(x, z) = zxx−1F1,1(x;x + 1;−z), a standard identity (See, for example, Andrewset al. (2000)). Rearranging, (57) is equivalent to

(59)e−θF2,2(1− λ, 1− λ; 2− λ, 2− λ;−θ)

F1,1(1− λ; 2− λ;−θ)F1,1(2− λ; 3− λ;−θ) < 1.

To establish (59) and hence prove that ds∗K/dλ > 0, it suffi ces to prove the following generallemma. Inequality (59) is the special case where a = 1− λ and x = θ.

Lemma 4. For any a ≥ 0 and any x > 0, we have

(60) e−xF2,2(a, a; a+ 1, a+ 1;−x) < F1,1(a; a+ 1;−x)F1,1(a+ 1; a+ 2;−x).

Proof. First, we use the following result found in Miller and Paris (2012) just after Eq.(5.3), obtained by specialization of 9.1 (34) in Luke (1969).

(61) F2,2(a, f ; b, c;−x) =

∞∑k=0

(a)k(c− f)k(b)k(c)k

xk

k!F1,1(a+ k; b+ k;−x)

32

Setting f = a and b = c = a+ 1 in (61), and using the fact that (1)k = k!,

F2,2(a, a; a+ 1, a+ 1;−x) =

∞∑k=0

(a)k(a+ 1)2

k

xkF1,1(a+ k; a+ 1 + k;−x)

Next, we apply Kummer’s first transformation, F1,1(y; z;−x) = e−xF1,1(z− y; z;x) to allF1,1 terms. (See, for example, Andrews et al. (2000), [Eq. 4.1.11]). Replacing F1,1(1; a +2;x) with its definition and cancelling the term e−2x from both sides, inequality (60) is

(62)∞∑k=0

(a)k xk

(a+ 1)2k

F1,1(1; a+ 1 + k;x) < F1,1(1; a+ 1;x)

∞∑k=0

xk

(a+ 2)k

Since all terms in both series are positive now, we can simply compare coeffi cients of likepowers of x. Inequality (62) holds provided that for all k ∈ N,

(63)(a)k

(a+ 1)2k

F1,1(1; a+ 1 + k;x) < F1,1(1; a+ 1;x)1

(a+ 2)k

First, it is straightforward to verify that the following holds:

(a)k(a+ 2)k(a+ 1)2

k

=a(a+ k + 1)

(a+ 1)(a+ k)≤ 1

Also, F1,1(1; a + 1 + k;x) < F1,1(1; a + 1;x) for all k ∈ N since ∂F1,1(a1;b1;x)

∂b1< 0. (See

Erdelyi, Magnus, Oberhettinger, and Tricomi (1953) for this derivative.)

33

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